Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455572 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,8)(9,10)(11,23)(12,27)(13,29,33,38)(14,21)(15,20)(16,22)(17,25)(18,26)(19,32,37,31)(24,35,28,34)(30,36,39,40);(1,2,4,5)(3,6,7,8)(9,11,15,26,16,12,14,25)(10,17,21,27,22,18,20,23)(13,24,31,39,33,28,32,30)(19,34,29,40,37,35,38,36);(3,7)(6,8)(9,12)(10,18)(11,16)(13,30)(14,25)(15,26)(17,22)(19,36)(20,23)(21,27)(24,32)(28,31)(29,35)(33,39)(34,38)(37,40);(1,3,4,7)(2,6,5,8)(9,13,16,33)(10,19,22,37)(11,24,12,28)(14,32,15,31)(17,34,18,35)(20,38,21,29)(23,36,27,40)(25,30,26,39)Polycyclic presentation:
2^9|a^2=fg,b^2=efgi,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374999);